Linear Functions vs Nonlinear Functions - Free Printable

It looks like you've uploaded a worksheet titled "Connect Graphs and Equations" with several problems related to identifying whether given graphs represent linear functions, matching equations to graphs, and understanding the characteristics of linear vs. nonlinear functions.

Let's go through the key parts of the worksheet and solve them step-by-step.

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🔍 Overview of the Worksheet

The worksheet has four main sections:

1. Connect Graphs and Equations – Match graphs to equations and determine if they are linear.
2. Connect Graphs and Equations (continued) – More graph-to-equation matching.
3. Explain – Conceptual questions about linear functions.
4. Answers – Partial answers provided.

We’ll walk through each section and explain how to solve it.

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## ✔ Section 1: Connect Graphs and Equations

Each problem gives a graph and asks:
- Write an equation.
- Is this a linear function? (Yes/No)

Let’s analyze each one based on typical patterns.

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Problem 1

Graph: A horizontal line at y = 3
✔ Equation: $ y = 3 $
✔ Is this linear? Yes
> *A horizontal line is linear because it's a straight line. It has a slope of 0.*

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Problem 2

Graph: A V-shaped graph (absolute value)
✔ Equation: $ y = |x| $
✘ Is this linear? No
> *Although it consists of two straight lines, the entire graph is not linear because it does not follow a single straight line. The function changes direction at x = 0.*

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Problem 3

Graph: A parabola opening upwards (U-shape)
✔ Equation: $ y = x^2 $ or similar
✘ Is this linear? No
> *Quadratic functions are not linear; they form curves.*

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Problem 4

Graph: A vertical line at x = 2
⚠️ Equation: $ x = 2 $
✘ Is this linear? No — but wait! This is tricky.

> Important Note: A vertical line is not a function (fails the vertical line test), so it cannot be a linear *function*. However, in some contexts, people might say it's "linear" as a relation. But strictly speaking, it is not a function, so the answer is no.

So:
✔ Equation: $ x = 2 $
✘ Is this a linear function? No

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## ✔ Section 2: Connect Graphs and Equations

Now we have more graphs.

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Problem 7

Graph: A straight line passing through origin with positive slope
✔ Equation: $ y = 4 \cdot x $
✔ Is this linear? Yes
> *This is a direct proportion, which is linear.*

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Problem 8

Graph: A curve that starts flat and increases steeply (exponential-like)
✔ Equation: $ y = 1.2^x $
✘ Is this linear? No
> *Exponential functions grow faster over time and are not straight lines.*

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Problem 9

Graph: A hyperbola (two branches)
✔ Equation: $ y = \frac{1}{x} $ when $ x \neq 0 $
✘ Is this linear? No
> *This is a rational function, not linear.*

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Problem 10

Graph: A downward-opening parabola
✔ Equation: $ y = -3 \cdot x^2 + 4 $
✘ Is this linear? No
> *Quadratic function → not linear.*

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Problem 11

Graph: A straight line with negative slope
✔ Equation: $ y = -3 \cdot x + 4 $
✔ Is this linear? Yes
> *Any equation of the form $ y = mx + b $ is linear.*

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Problem 12

Graph: A straight line with positive slope, not through origin
✔ Equation: $ y = x + 1 $
✔ Is this linear? Yes
> *Again, standard linear form.*

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## ✔ Section 3: Connect Graphs and Equations

More graphs and equations.

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Problem 13

Graph: Straight line through origin, positive slope
✔ Equation: $ y = x $
✔ Linear? Yes

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Problem 14

Graph: Same as above, but steeper
✔ Equation: $ y = 2 \cdot x $
✔ Linear? Yes

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Problem 15

Graph: U-shaped parabola (vertex at origin)
✔ Equation: $ y = x^2 $
✘ Linear? No

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## ✔ Section 4: Explain

Question 19: What do linear functions look like in a coordinate plane?

👉 Answer: Linear functions appear as straight lines. They have constant rate of change (slope).

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Question 20: Can you think of any situation when a straight line is not a function?

👉 Answer: Yes — a vertical line (e.g., $ x = 3 $) is not a function because it fails the vertical line test (one input maps to multiple outputs). So even though it's a straight line, it's not a function.

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Question 21: Give three examples of equations of linear functions.

👉 Examples:
1. $ y = 2x + 1 $
2. $ y = -x + 5 $
3. $ y = 0.5x $

> All are in the form $ y = mx + b $, where $ m $ and $ b $ are constants.

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Question 22: Give two examples of equations of nonlinear functions.

👉 Examples:
1. $ y = x^2 $
2. $ y = \frac{1}{x} $

> These are nonlinear because their graphs are not straight lines.

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## ✔ Summary Table: Key Takeaways

| Graph Type | Equation Example | Linear Function? | Notes |
|----------------------|-------------------------|------------------|-------|
| Horizontal line | $ y = 3 $ | ✔ Yes | Constant function |
| Vertical line | $ x = 2 $ | ✘ No | Not a function |
| V-shape | $ y = |x| $ | ✘ No | Absolute value |
| Parabola | $ y = x^2 $ | ✘ No | Quadratic |
| Exponential | $ y = 1.2^x $ | ✘ No | Growth curve |
| Hyperbola | $ y = \frac{1}{x} $ | ✘ No | Rational |
| Straight line | $ y = 2x + 3 $ | ✔ Yes | Linear |

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## ✔ Final Answer: How to Solve These Problems

To solve such problems:

1. Identify the shape of the graph:
- Straight line → likely linear
- Curve (parabola, exponential, etc.) → nonlinear
- Vertical line → not a function

2. Match to known equations:
- Use standard forms:
- Linear: $ y = mx + b $
- Quadratic: $ y = ax^2 + bx + c $
- Absolute value: $ y = |x| $
- Reciprocal: $ y = \frac{1}{x} $

3. Check if it's a function:
- Use the vertical line test.
- If any vertical line crosses the graph more than once → not a function.

4. Answer “Is this a linear function?”:
- Only say yes if it's a straight line and passes the vertical line test.

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## 📝 Conclusion

You can now confidently match graphs to equations and classify them as linear or nonlinear using these rules.

If you want me to help fill in the blank spaces in the worksheet (like writing equations for specific graphs), just let me know which ones you're unsure about!

Would you like a printable version or a completed worksheet PDF?